L(s) = 1 | + 2-s + 3-s + 4-s − 0.176·5-s + 6-s − 2.27·7-s + 8-s + 9-s − 0.176·10-s − 2.71·11-s + 12-s + 13-s − 2.27·14-s − 0.176·15-s + 16-s − 1.27·17-s + 18-s + 3.21·19-s − 0.176·20-s − 2.27·21-s − 2.71·22-s − 5.17·23-s + 24-s − 4.96·25-s + 26-s + 27-s − 2.27·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0791·5-s + 0.408·6-s − 0.858·7-s + 0.353·8-s + 0.333·9-s − 0.0559·10-s − 0.818·11-s + 0.288·12-s + 0.277·13-s − 0.607·14-s − 0.0456·15-s + 0.250·16-s − 0.309·17-s + 0.235·18-s + 0.737·19-s − 0.0395·20-s − 0.495·21-s − 0.578·22-s − 1.07·23-s + 0.204·24-s − 0.993·25-s + 0.196·26-s + 0.192·27-s − 0.429·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 0.176T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 + 0.285T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 - 1.27T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 6.63T + 79T^{2} \) |
| 83 | \( 1 - 0.0726T + 83T^{2} \) |
| 89 | \( 1 + 0.653T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60634763465458385651443864058, −6.61279166461756702337755271156, −6.17968480061542349986505979205, −5.36097167598233759216677538146, −4.57679915389290463219580834367, −3.80501074108590618057500146047, −3.14049368476679320259447966174, −2.52121476722538677395775615408, −1.53533248934244855582213772571, 0,
1.53533248934244855582213772571, 2.52121476722538677395775615408, 3.14049368476679320259447966174, 3.80501074108590618057500146047, 4.57679915389290463219580834367, 5.36097167598233759216677538146, 6.17968480061542349986505979205, 6.61279166461756702337755271156, 7.60634763465458385651443864058